The axion finally starts the second stage of oscillations around one of the \({\cal N}\)  zero-temperature minima at random.
Thus, the \(1/{\cal N}\)  probability of the \(Z_{\cal N}\)  axion scenario [1]}, [2]} to solve the strong CP problem (i.e. the final \(\theta \) -parameter to corresponding the CP-conserving point \(\theta =0\) ) will be maintained when accounting for DM.
